
2D Poisson equation: The Poisson's equation, shown in Eq. REF , is very popular in engineering simulations, for example, chip temperature prediction, pressure equation in fluids etc.
\(\nabla ^2 u = f\) 
where, \(u\)  is the solution variable and \(f\)  is the source term. This PDE is solved on a 1024x1024 grid to resolve the high-frequency features of the source term. The source, \(f\) , is sampled from a Gaussian mixture model shown in Eq. REF .
\(\sum _{j=0}^{1024}\sum _{j=0}^{1024} f_{i,j} = \sum _{j=0}^{1024}\sum _{j=0}^{1024}\sum _{k=0}^{30} A_k\exp \left( -\left(\frac{x-\mu _{x,k}}{\sigma _{x,k}}\right)^2 - \left(\frac{y-\mu _{y,k}}{\sigma _{y,k}}\right)^2 \right)\) 
where, \(x, y\)  correspond to the grid coordinates. \(A_k\)  randomly assumes either 0 or 1 to vary the number of active Gaussians in the model. \(\mu _x, \mu _y\)  and \(\sigma _x, \sigma _y\)  are the mean and standard deviations of Gaussians in \(x\)  and \(y\)  directions, respectively. The means and standard deviations vary randomly between 0 to 1 and \(0.001\)  to \(0.01\) , respectively. The smaller magnitude of standard deviation results in hot spots that require highly-resolved grids. In this case, 256 training and 100 testing solutions are generated using Ansys Fluent.

2D non-linear coupled Poisson equation: The coupled non-linear Poisson's equation is shown below in Eq. REF . \(u, v\)  are the solution variables and \(f\)  is the source term, similar to the description in Eq. REF . This PDE has applications in reactive flow simulations.
\(\nabla ^2 u = f - u^2; \quad \quad \nabla ^2 v = \frac{1}{u^2+\epsilon } - v^2\) 
The data generation is similar to Experiment 2 with a slight difference that the source term is less stiff with a standard deviation that varies between \(0.005\)  to \(0.05\) .

3D Reynolds-Averaged Navier-Stokes external flow: This use case consists of a 3-D channel flow with resolution 304x64x64 at high Reynolds number over arbitrarily shaped objects. The corresponding PDEs are presented in supplementary materials section C.3 or may be referred from [1]}. The characteristics of flow generated on the downstream has important applications in the design of automobiles and airplanes. In this case, we use 5 primitive 3-D geometries namely, cylinder, cuboid, trapezoid, airfoil wing and wedge and their random combinations and rotational augmentations to create 150 training geometries and 50 testing geometries, which are solved with Ansys Fluent to generate the data. Out-of-distribution testing is carried out on simplistic automobile geometries as shown in Figure REF .

3D chip cooling with Natural convection: In this case, we extend the complexity of Experiments \(1, 2\)  and 3 to a industrial 3-D case of chip cooling with natural convection. In this problem, the computational grid (\(128^3\)  resolution) consists of a chip that is subjected to the powermap specified by Eq. REF . The heating of the chip results in generation of velocity in the fluid and at steady-state, a balance is achieved. We use Ansys Fluent to generate 300 training and 100 testing solutions for velocity, pressure, temperature for arbitrary sources.

